I've heard that coffee grounds are a good "green"(nitrogen rich) to put in your compost, but I was wondering if I could throw the filter in there as well? They are the standard "accordion side" white ones.
The paper ones? Sure - they're just like blotting paper and break down quickly. The usual things that can make some papers questionable should not be present (plastic coatings, metal and/or oil based inks) - or if they are present, you need to quickly change your filter brand!
What I would probably do if possible, is to try and tear the paper a little so that it all mixes up more easily and quickly.
Yes, you can add coffee grounds with their filters to your compost pile. Because they are wet, they decompose fairly quickly. The filters may dry out you if leave them on top of the pile in dry weather. Keep it inside the pile and keep it moist. Also, worms compost the grounds and filters very fast. I have a five tray worm factory and I just drop the whole filter with the grounds in and close the lid, and my army of five thousand redworms eat it in less than a week. I found filters to be easy to compost in almost any composter. If you have a two week ComposTumbler you will want to shred them before you add them, to keep it even.
I don't mind they don't break down fast, I consider it a mulch for the soil and or weed discourager. I have my colleagues use a plastic bin to save all coffee grounds with their filters and I recycle them to my garden. You can imagine the amount of filters that go into my garden. I haven't seen a problem at all with them. It's true if the filters are exposed to air they do not decompose unless covered in soil.